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On the Solutions of a Stochastic Control System. Ii* SIAM J. CONTROL Vol. 13, No. 5, August 1975 ON THE SOLUTIONS OF A STOCHASTIC CONTROL SYSTEM. II* TYRONE DUNCAN'}" AND PRAVIN VARAIYA: Abstract. This paper presents generalizations of the work in [1], [2] to include controlled stochastic processes which take values in a certain class of Fr6chet spaces. The crucial result is an extension of Girsanov's technique for defining solutions of stochastic differential equations by an absolutely continuous transformation of measures. The result is used to prove existence results for stochastic control problems and for a class of two-person zero sum games. 1. Introduction and summary. Consider a controlled stochastic process represented by the stochastic differential equation dX(t) f(t, X, u(t, X)) dt + dB(t), [0, 1], where B(t) is a Fr6chet-valued Brownian motion, X(t) is the state process. The drift f, and the control u, depend at any time on the past of the state process, {X(s), s __< t}. For the case where the Fr6chet space is Rn, a satisfactory theory dealing with the problem of existence of solutions of the differential equation and existence of optimal control laws is now available [1], [2]. A crucial building block in this theory consists in defining a solution of the differential equation via an absolutely continuous transformation of measures. Each control thereby de- fines a solution characterized by its (unique) probability law which is absolutely continuous with respect to Wiener measure. Thus the influence of a control law upon the system is captured in the Radon-Nikodym derivative of the resulting probability law with respect to Wiener measure. Questions dealing with the existence of an optimal control can then be converted into questions about the compactness (in an appropriate sense) of the set of Radon-Nikodym derivatives. The measure transformation technique mentioned above is due originally to Girsanov 16]. This paper deals with these same questions for the case where the state space is infinite-dimensional. The problem of characterizing Brownian motion in infinite-dimensional spaces is a difficult one and has been resolved for certain Fr6chet spaces only. This is described in the next section, where some additional properties of such Brownian motion as sample continuity and stochastic inte- gration are established also.. In 3 the result of Girsanov is extended to cover the differential equation under consideration. Once this has been achieved the tech- niques of [1], [2] apply without change and the existence of optimal controls and saddle points follows easily. This is sketched in 4. 2. Preliminaries for defining Brownian motion. To define probability measures on Fr6chet spaces the results of Dudley-Feldman-LeCam [3] are used. The Received by the editors September 25, 1973, and in revised form August 29, 1974. f Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, New York 11790. This research was supported in part by the National Science Foundation under Grants GK-24151 and GK-32136. :I: Department of Electrical Engineering and Computer Science and Electronics Research Lab- oratory, University of California, Berkeley, California 94720. This research was supported in part by Downloaded 09/10/14 to 129.237.46.100. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php the National Science Foundation under Grant GK-10656X3. 1077 1078 TYRONE DUNCAN AND PRAVIN VARAIYA latter are generalizations of the work of Gross [4]. While the generality of the immediately following discussion is not used subsequently, it does serve to indicate some directions along which the results reported here can be further pursued. Some notation and definitions are introduced first. For a locally convex Huasdorff topological (real) vector space X let X* denote its topological dual. A dual system or duality over the reals consists of two vector spaces X, Y and a bilinear form (.,. X Y--* R that separates points for both X and Y. For a pair X, Y in duality, let FD(X) denote the collection of all finite-dimensional subspaces of X. For G c X, let///(Y, G) be the a-algebra of Y generated by the sets (y: (x, y) B}, where x G, B is a Borel set in R. Thus /J/(Y, G) is the smallest r-algebra on Y for which each x G, regarded as a function on Y, is measurable. Finally let (g(Y, X) U (/J/(Y, G)IG FD(x). A cylinder set measure on Y is any nonnegative, finitely additive set function m on (g(Y, X) with m(Y) such that m is countably additive on '(Y, G) for each G FD(X). The following notion of a measurable seminorm given in [3] is crucial to the analysis. DEFINITION 1. Given a duality X, Y, (.,. , and a cylinder set measure m on Y, a seminorm I" on Y is said to be m-measurable if for each e > 0 there is a G FD(Y) such that if F FD(X) and F d_ G (i.e., (x, y 0 for x e F, y G), then m{y:ly- F+/-I _-< e} _-> 1-e or equivalently m*{y:iy- Gl<e} l-e, where m* is the outer measure on cg(y, X) induced by m. The next result follows from [3, Thm. 2]. FUNDAMENTAL THEOREM. Let I" be a Mackey-continuous seminorm on Y and suppose that it is m-measurable. Then the cylinder set measure induced by m on the Banach space Y/I" obtained from Y via the seminorm ]. extends to a regular Borel measure. From here on the discussion is specialized to a fixed, separable Hilbert space H. It is assumed that there is given for each R+ a cylinder set measure Pt on H such that pt is a canonical normal distribution on H with variance param- eter (see [4] or [5]). It is also assumed that there is given an increasing family of Mackey-continuous seminorms I" I, j 1, 2,..., on H. Let F be the Fr6chet space obtained from H with respect to the topology defined by the seminorms I" I by completion modulo the intersection of their null spaces (which without loss of generality is assumed to equal {0}). As a corollary to the Fundamental Theorem it is proved in [3, Cor. 2.1] that each Pt, tR+, extends to a regular Borel measure, denoted #t, on F. For future reference note that H c F, F* H* This means that if C e (H, is of the form C P- (E), where P is the orthogonal projection of H onto a subspace L FD(H) and E is a Borel subset of L, then pt(C) (2t)-"/Z feexp (- lxl dx, norm on its inner product. Downloaded 09/10/14 to 129.237.46.100. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php where n is the dimension of L, and ]. lu denotes the H induced by STOCHASTIC CONTROL SYSTEM 1079 H, and define the maps i'H F, j'F* H*, as the canonical injections. Finally let N(F) denote the Borel sets of F. Throughout, H and F denote the spaces introduced here. The collection #t, e R/, will be used now to define a Brownian motion with values in F. For eacl e R/ let F be a copy of F. For each finite collection t < < t,, let tl,'",trt be the measure on (]-[7= Ft" HT=l (Ft')) defined by fAn d#tl'""t" d#,._t._,(x,,) dl.tt=_t,(x2) dYt,(Xl) for Ai (F"), 1, ..., n. Since sets of the form l--I'= Ai generate the product r-algebra ]-I'-1 (F") the measure #,1,...,,, is defined. To show that the family of measures #,1,...,,, is a projective system, it is necessary to verify consistency. Since F is separable in the topology determined by the countable seminorms I. j, the measure #,1,...,,, is determined by sets of the form Ai (I P)F @ Fi, where P is an orthogonal projection with finite dimensional range and F is a Borel subset of PF. Let A1, "'", A, be such sets, and suppose Aj F. Then d'''''"= dl'tt"-t"-'(xn)'"d#t*(x1) fA fA fA fA 2--Xl y.- 1Xi @,._,._ ,(x,) dp,._,._ ..dp,,+ =-t,+ dpt,+ ,-t,_,(x) [l,',[j l[j+ 1,",n j-1 j+l In the above p is the measure of a Brownian motion with values in the finite- dimensional space PF and so the third equality above follows from the Markov property of such a process. Since each of the measures #,, R/, is a regular Borel measure, the projective system of measures admits a projective limit [6, p. 49]. The projective limit thus obtained is denoted (f2, ff, P). Evidently, (f2, )= 1-[,e+ (U, (F')). It will be assumed that (f2, if, P) is complete. For each let X denote the F-valued evaluation map on f2 at t. Let o be the smallest completed a-algebra with respect to which X is measurable for s =< t. Then (Xt, t, P)te is a Brownian motion as defined below. DEFINITION 2. A stochastic process (Xt, ,, P)tzR+ (or simply (X,, P)tzR or (X,) if there is no ambiguity) is said to be a Brownian motion with values in the Frdchet space F induced from a family of canonical normal distributions on a Hilbert space H, if for each e F* (the topological dual of F) the real-valued process ((1, X,), ,P)tzR+ is a real-valued Brownian motion with E<l, Xt> 2 tljll2n.
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